Computational Materials Science 24 (2002) 421–429
www.elsevier.com/locate/commatsci
Pressure-transmitting boundary conditions for
molecular-dynamics simulations
Carsten Sch€
afer a, Herbert M. Urbassek a,*, Leonid V. Zhigilei b,
Barbara J. Garrison c
a
Fachbereich Physik, Universit€at Kaiserslautern, Erwin-Schr€odinger-Straße, D-67663 Kaiserslautern, Germany
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903, USA
Department of Chemistry, 152 Davey Laboratory, Pennsylvania State University, University Park, PA 16802, USA
b
c
Received 10 October 2001; accepted 20 November 2001
Abstract
A scheme for establishing boundary conditions in molecular-dynamics simulations that prevent pressure wave
reflection out of the simulation volume is formulated. The algorithm is easily implemented for a one-dimensional
geometry. Its efficiency is tested for compressive waves in Cu. 2002 Elsevier Science B.V. All rights reserved.
PACS: 02.70.Ns
1. Introduction
Molecular-dynamics calculations are generally
used to simulate macroscopic systems containing a
large number (in the order of Avogadro’s number,
1023 ) of atoms. By necessity, the simulation concentrates on a small part of the system, on the
order of several hundreds [1] to a maximum [2]
of nowadays 5 109 atoms. This reduction in system size is possible, if convenient boundary conditions are applied to the simulation volume; these
boundary conditions are chosen to mimic the re-
*
Corresponding author.
E-mail address: urbassek@rhrk.uni-kl.de (H.M. Urbassek).
URL: http://www.physik.uni-kl.de/urbassek.
sponse of the surrounding material to the processes occurring in the simulation volume itself.
The boundary conditions applicable to simulations in thermodynamic equilibrium (static response) have been well studied in the past [3–6].
However, nowadays, often non-equilibrium processes are to be simulated. Then the formulation
of the appropriate boundary conditions poses a
larger challenge, as they need to incorporate the
dynamic response of the surroundings to the processes occurring in the simulation volume. Thus,
for example, if heat is produced in the simulation
volume, the boundaries have to dispense of it in
a realistic way, mimicking the natural heat conduction of the solid. This can be achieved using
so-called energy-dissipating boundary conditions
[7,8]. This problem arises, e.g., when energetic
processes, such as ion or laser irradiation of a solid
0927-0256/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 6 3 - 4
422
C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
surface, are simulated [7,9]. Analogously, stress
produced in the simulation volume has to be relaxed at the boundaries in a way compatible with
the elastic properties of the solid. Again, this is
quite a commonly encountered phenomenon in
irradiation or impact simulations [10,11].
The latter class of boundary conditions is more
difficult to formulate. In fact, a number of schemes
have been published recently to solve this problem
that are based on a generalized Langevin approach
[12–14], in which the reaction of the surrounding is
incorporated into the dynamical equations of the
simulation volume via (delayed) forces [15,16]. It
turns out, however, that the calculation of these
boundary conditions is rather time consuming.
For this reason, we propose in this paper a simple
phenomenological scheme to handle pressure relaxation and pressure wave transmission, which
is easy to implement and adds virtually no
extra computational costs to the simulation. This
scheme was introduced in Ref. [17], and used repeatedly for applications in laser ablation of solids
[11,18,19]. We shall validate the scheme for a metallic system by investigating its ability to transmit
pressure waves.
2. Formulation of boundary conditions
2.1. Passage of a pressure wave through matter
In order to set the stage, we remind the reader
of the basic properties of a pressure wave in a
solid. Fig. 1 exemplifies the conditions in a material through which a pressure wave passes. The
figure was taken from a molecular-dynamics simulation of Cu; the wave was started by applying a
sudden force perpendicular to the surface (for details see Section 3 below). The quantities discussed
below denote local averages; they were determined
in the simulation as averages over a distance of
, the cutoff radius of our potential.
rc ¼ 6:2 A
/
The pressure wave runs with a velocity of 55 A
ps, which is 25% above the speed of sound, which is
/ps for a longitudinal wave in Cu in the
c0 ¼ 44 A
[1 0 0] direction; it is hence a (moderately) weak
pressure wave [20]. In the wave itself, the density n
is increased by around 10% above the nominal
3 . In the region of
crystal density n0 ¼ 0:085 A
, roughly
enhanced density––between 27 and 35 A
––the atom velocity u is non-vanishing; its values,
however, are small compared to c0 . The pressure p,
however, starts to become already strong in the
leading part of the wave, before atoms have
reached sizable velocities. This is in contrast to a
well-known relationship [21], which has pressure to
be proportional to the atom velocity
p ¼ mn0 c0 u ¼ Zu;
ð1Þ
where m is the atom mass, and Z ¼ mn0 c0 is called
the impedance of the system; in our case, Z ¼ 3:96
ps) ¼ 24:7 meVps/A
4 . Inspection of Fig. 1b
GPa/(A
and c shows that this simple proportionality does
not hold on the microscopic (atomistic) scale. It is
the purpose of the following section to show that a
closely related proportionality does indeed hold
for a well-defined part of the forces acting on an
atom, viz. Eq. (4) below.
2.2. Boundary forces
As soon as such a pressure wave reaches the
boundary of the simulation volume, it will be reflected. This happens for both types of boundary
conditions commonly applied, namely free and
fixed boundaries. In order to prevent reflection,
one has to mimic the action of the surrounding
medium on the atoms close to the boundary. We
do this in the following way: we define as boundary zone the part of the simulation volume adjacent to the boundary (cf. Fig. 2); its width Dz must
be (slightly) larger than rc . Atoms in the rest of the
simulation volume (the MD zone) obey the original MD equations of motion with forces stemming
from the neighboring atoms. Thus in Fig. 2, atom i
experiences a force F stemming from all atoms j
surrounding it. Let us write F as
F ¼ Ftop þ Fbottom ;
ð2Þ
where Ftop is the force originating from all atoms
above i (i.e., away from the boundary), and Fbottom
is the force originating from all atoms below it (i.e.,
closer to the boundary). Here and in the following
we shall be interested only in the force components
C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
423
Fig. 1. Molecular-dynamics results of a pressure wave travelling through a Cu crystallite in [1 0 0] direction. The wave was excited
for 11 fs on each surface atom in the direction perpendicular to the surface. The data were taken
by applying a force K ¼ 7:5 eV/A
at 0.6 ps after pulse excitation. (a) Density n, (b) atom velocity u, (c) pressure p, (d) force F split into top and bottom part,
Ftop and Fbottom , respectively, according to Eq. (2), cf. Fig. 2, thick line: Eq. (4), (e) potential and kinetic energy, Epot and Ekin , respectively.
perpendicular to the boundary; hence we can abstain from using vector notation.
Fig. 1d shows F and its two components Ftop
and Fbottom in a pressure pulse. We observe that
Ftop and Fbottom have a simple unimodal shape,
while the total force F shows the more complex
structure typical of a pressure pulse, consisting
of an accelerating and a decelerating part.
While using interatomic interaction to determine
both Ftop and Fbottom in the molecular-dynamics
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C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
Fig. 2. Schematics of the simulation volume, partitioned into a
boundary zone, and the MD zone proper. Forces on an atom i
are shown. Forces are split into Ftop (arising from atoms further
away from the boundary) and Fbottom (arising from atoms closer
to the boundary).
zone, in the boundary zone we model Fbottom by a
simple expression Fbc such that atoms in the
boundary zone experience a total force
F ¼ Ftop þ Fbc ;
ð3Þ
in analogy to Eq. (2). Ftop is calculated from the
interatomic potentials in the same way as in the
MD zone. The boundary force, Fbc , is directed
perpendicular to the boundary. Its magnitude is
determined as
Fbc ¼ F0 au:
ð4Þ
Here, F0 is a static contribution, which is present
even if atoms do not move. It is needed to keep the
crystal in equilibrium, to counter the force of the
top atoms and to eliminate the surface tension
forces acting on the atoms in the boundary region.
The second part of the force, Eq. (4) is proportional to the velocity u of the atoms in the
boundary. For the purposes of this paper––short
pulses and low ambient temperatures––it is appropriate to take u as the velocity of the individual
boundary atom. More generally, this procedure
would induce an artificial cooling of the boundary,
and u should be taken as the average velocity
(center-of-mass velocity) of all atoms in the
boundary [17]. According to Eq. (1), we choose the
proportionality constant a as
a ¼ ZA ¼ mn0 c0 A;
ð5Þ
where A is the cross-sectional area of an atom.
Thus, Eq. (4) makes sure that the atoms in the
boundary zone are subject to the pressure, Eq. (1),
that they need to feel under the passage of a
pressure pulse. By prescribing the correct impedance to the boundary we prevent acoustic waves
from reflecting. We shall hence refer to a as the
impedance coefficient in this paper. Note that
f ¼ a=m may be interpreted as a friction coeffi2=3
cient in Eq. (4). Since A is of the order of n0 , f
is ffiffiffiffiffiffiffiffiffiffiffi
proportional
to the Debye frequency xD ¼
ffi
p
3
6p2 n0 c, where c is an appropriately averaged
speed of sound. A friction of fAD ¼ pxD =6 was
indeed derived by Adelman and Doll [12] in their
generalized Langevin approach by using a Brownian approximation.
We emphasize that the boundary zone must be
determined dynamically during the simulation, to
keep it a width Dz above the bottommost atom
layer; its absolute position and even the identity
of its atoms can change during the simulation.
2.3. Determination of parameters
In the following, we shall discuss the determination of the parameters necessary to implement
the boundary conditions. We shall do this specifically for a many-body potential as it is adequate
for a metal, such as the EAM potential [22]. The
implementation for two-body potentials can be
viewed as a special case of this more general type
of potentials by setting the many-body embedding
function equal to zero. We shall establish the parameters here for the many-body potential defined
in Ref. [23].
For many-body forces, the partitioning of the
(well-defined) total force into Ftop and Fbottom needs
an explication. We proceed as follows: let us write
the total potential energy of the crystal as
X
X
Etot ¼
Uðrij Þ þ
Gðqi Þ;
ð6Þ
i<j
with
qi ¼
X
gðrij Þ:
i
ð7Þ
j6¼i
Here U is a pair potential, G the embedding
function, and g the atomic contribution to qi , the
C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
425
so-called electron density. Then the total force on
atom i is
X
Fij ;
ð8Þ
Fi ¼
j6¼i
where
Fij ¼ fU0 ðrij Þ þ g0 ðrij Þ½G0 ðqi Þ þ G0 ðqj Þgeij :
ð9Þ
Here the dash denotes differentiation with respect to the argument and eij is the unit vector
pointing from atom j to atom i. Fij may be interpreted as the force contributed by atom j to the
total force on atom i; this makes the partitioning
of Fi into Ftop and Fbottom by the position of atom
j with respect to atom i obvious.
We note that these formulae allow us to calculate F0 , and hence the static part of the boundary
force, analytically. Alternatively, F0 can be conveniently determined from a static simulation of
the crystal in equilibrium (pressure p ¼ 0). Both
.
procedures yield F0 ¼ 0:302 eV/A
The dynamical part of the boundary force is
governed by the parameter a. Using Eq. (5), it
could be determined from the impedance of the
crystal and the atomic cross-section, which (for the
(1 0 0) surface) may be set equal to A ¼ a2 =2 ¼
2 . This gives a value of atheor ¼ 0:162 eVps/
6:53 A
2
, corresponding to a friction coefficient f ¼ 2c0 =
A
a ¼ 24:4/ps.
As an alternative procedure, we may determine
a by performing a dynamical simulation of a
pressure wave travelling through a crystal. The
dependence of Fbottom on the velocity u in a region
far away from all boundaries can thus be determined. The result of such a simulation is shown in
Fig. 3a. It is seen that Fbottom correlates quite well
with the individual velocity of each atom. A linear
2 , in quite good
fit gives asimul ¼ 0:193 eVps/A
agreement with the estimate atheor given above. We
plotted Fbc according to Eq. (4) with parameters as
discussed above also in Fig. 1d. Good agreement
with the simulated values of Fbottom is observed.
Note, however, that in the leading part of the
pulse, Fbc overestimates Fbottom , while it underestimates it in the trailing part.
We note that we performed the same fit for a
soft material, condensed Ar. The results, obtained
for a Lennard–Jones pair potential [24], are shown
Fig. 3. Dependence of the force Fbottom acting on individual
atoms vs their velocity u. Simulation data for a pressure wave
travelling in (a) Cu and (b) Ar. Line: fit to Eq. (4).
in Fig. 3b. Here for negative velocities, a bi-valued
result is obtained. This sort of hysteresis behavior
is analogous––but more pronounced––to the behavior found for Cu and discussed above: in the
leading (trailing) part of the wave smaller (larger)
forces than predicted by Eq. (4) occur. A linear fit
is then useful only as an average or for one part of
the hysteresis loop.
3. Validation
We shall use in this paper Cu as a model system
to exemplify and test our scheme. The moleculardynamics simulation employs the many-body potential defined in Ref. [23]; this is of the so-called
embedded-atom [22] or tight-binding [25] form.
The simulation employs a Cu crystallite at 0 K
with a (1 0 0) surface and a cross-sectional area of
2
is the Cu lattice constant.
ð6aÞ , where a ¼ 3:615 A
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C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
While the surface is left free, lateral periodic
boundary conditions are applied at all four sides.
At the bottom of the simulation volume, we shall
employ the pressure-transmitting boundary conditions, which are the subject of this paper.
A convenient way to test our scheme is to
measure to which degree the boundary conditions
are able to prevent pressure waves from reflection.
To this end we prepare a pressure wave by applying a force K for a short period of time (here:
Dt ¼ 11 fs) on each surface atom in the direction
perpendicular to the surface. Fig. 1 showed the
form of the resulting pressure wave in the material
.
for a value of K ¼ 7:5 eV/A
In order to quantify wave reflection, we introduce the reflection coefficient
R ¼ Erefl =Ein ;
ð10Þ
where Ein is the (constant) kinetic energy in the
wave before reaching the boundary and Erefl is the
kinetic energy in the crystal after wave reflection.
We measured Erefl some time after wave reflection,
when it had become constant.
Let us first test whether the parameters F0 and a
appearing in the boundary force, Eq. (4), were
optimal in preventing wave reflection. For this
,
test, we fixed the wave strength to K ¼ 7:5 eV/A
and let the wave interact with a boundary with
varying values of F0 and a in the boundary force.
Fig. 4 displays the variation of the reflection coefficient with F0 and a. The optimum reflectivity
Ropt reaches a value of only 3.15%, which is sufficiently small for many applications. Note that our
pressure wave is very sharp; as was shown in Ref.
[15], it can be assumed that the reflection of waves
with less steep boundaries will be even more reduced than the sharp wave studied by us.
Ropt occurs for the value of F0 determined in
Section 2.3; however, the corresponding value of
2 is somewhat larger than that
a ¼ 0:21 eVps/A
found in Section 2.3 from Fig. 3, asimul ¼ 0:193
2 . The reason for this increase in the optieVps/A
mum value of a may lie in the strength of our
pressure pulse. We therefore performed another
determination of F0 and a for a really weak wave,
. Then the
excited with a force K ¼ 0:75 eV/A
parameter dependence is even more pronounced.
Furthermore, the optimum value of a has de-
Fig. 4. Reflection coefficient R of a weak pressure wave––ex and (b) 0.75 eV/A
––vs the
cited by a force (a) 7.5 eV/A
boundary impedance coefficient a. The curve parameter is the
constant contribution F0 to the boundary force.
2 , i.e.,
creased from 0.21 to a value of 0.19 eVps/A
the value determined from Fig. 3 in Section 2.3.
This is due to the fact that for the weak wave, the
wave speed is identical to the speed of sound c0 .
We note that also the scatter in the Fbottom vs u
correlation is reduced (not shown).
Note that R is not very sensitive to changes of
the impedance coefficient a in the order of 10%;
this is helpful, since the determination of a occurred either by the determination of crystal parameters like c0 , n0 , A, which may not be known to
an accuracy better than 10 % (note that in particular the wave speed may increase with wave
strength!); or by a fitting procedure as displayed in
Fig. 3, which again is subject to some error. On the
other hand, the dependence of R on F0 is quite
pronounced, and changes of F0 by a few % change
R distinctly. As outlined above, however, F0 can be
safely determined with high accuracy analytically
or by a static simulation.
C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
427
• We find pressure wave reflection to be suppressed by our method to a level of 3–4% which
is adequate for many applications.
Acknowledgements
The authors are grateful to M. Moseler and G.
Williams for discussions. They thank the computer
center RHRK, University of Kaiserslautern, for
making available computer time for this study.
Fig. 5. Reflection coefficient R vs strength K exciting the
pressure wave. Parameters F0 and a as determined in Section
2.3.
Fig. 5 shows how R varies with the strength of
the pressure wave. Here the optimum value of
2 as determined in Section 2.3
asimul ¼ 0:193 eVps/A
has been kept. In particular that we did not optimize a for each value of the force K. Note that R
is between 3% and 4% for a wide range of wave
strengths, and only increases towards very weak
waves. The reason hereto lies in the fact that these
quite weak waves become very delocalized during
the simulation, and look more like an acoustic
wave rather than a pressure pulse. These longer
wave trains transmit less efficiently through the
boundary zone than a well localized pulse. Thus,
the boundary impedance seems to work fine even
for stronger waves, where the wave speed is definitely larger than c0 .
4. Summary
• We formulated a conceptually simple and easyto-implement method which allows to minimize
pressure wave reflection at the boundary of a
molecular-dynamics simulation volume.
• The method is easily adapted to a variety of materials, including those with many-body interaction forces.
• Two parameters need to be determined for implementing the boundary force. We discuss several ways to determine these, and measure the
sensitivity of the method to the accuracy of
choosing the parameters.
Appendix A. Microscopic processes during pulse
passage
In this appendix, we shall study how the kinetic,
potential and total energies summed over all atoms
in the crystal change when a pressure pulse impinges on the boundary zone. Throughout this
appendix, we shall study a pulse excited with a
. Let us first study (Fig. 1e) the
force of 7.5 eV/A
spatially resolved energy content of the pressure
pulse displayed in Fig. 1. Epot has been set to zero
for a crystal in equilibrium. We observe again the
rather pronounced sharp front, followed by a
rather long, but considerably weaker tail. Note that
the front part of the pulse has negative potential
energy. This is a feature common to all realistic
potentials stretching out further than the nearestneighbor distance: atoms in front of the pulse feel
approaching atoms, and hence the bonding (attraction) increases. Only when nearest-neighbor
atoms come close, does repulsion set in, and the
potential energy becomes positive.
As a reference case, in Fig. 6, we display the
temporal changes of the kinetic and potential energies in a crystal of 30 ML length when the
pressure pulse impinges on a free surface. The
pressure pulse needs about 0.1 ps to equilibrate
potential and kinetic energy. Note that Ekin > Epot ,
since the pulse is strong enough that the crystal
forces become anharmonic. At 1 ps, the pulse is
reflected: at the free surface the crystal almost
completely relaxes ðEpot ¼ 0Þ while all energy is
converted to kinetic energy. The pulse shape
changes upon reflection; hence the oscillations
visible after reflection.
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C. Sch€afer et al. / Computational Materials Science 24 (2002) 421–429
Fig. 6. Molecular-dynamics results of a pressure wave travelling through a 30 ML Cu crystallite in [1 0 0] direction, cf. Fig.
1. Time dependence of potential, kinetic and total energy (Epot ,
Ekin , and Etot , respectively) summed over all atoms in the
crystal. Free boundary conditions.
Fig. 7. Molecular-dynamics results of a pressure wave travelling through a 60 ML Cu crystallite in [1 0 0] direction, cf. Fig.
1. Time dependence of potential, kinetic and total energy (Epot ,
Ekin , and Etot , respectively) summed over the top 30 ML in the
crystal.
Fig. 7 shows pulse transmission through an
‘ideal boundary’: this was realized by using a large
crystal (60 ML) and measuring the energies only in
the first 30 ML. Upon passage the kinetic energy
drops very sharply (within 50 fs) to about 5% of its
initial value. The later decrease to 0% is quite slow,
and takes more than 1 ps; this is due to the pulse
tail shown in Fig. 1e. The potential energy shows
somewhat more structure: immediately before
pulse passage it increases; this occurs when the
front part of the pulse possessing negative potential energy (Fig. 1e) has passed the ideal boundary.
Immediately after passage, it decreases (even
Fig. 8. Same as Fig. 6, but for pressure-transmitting boundary
conditions.
below 0!); this is due to the oscillatory nature of
the pulse visible in Fig. 1e. Note that the total
energy within the first 30 ML slightly increases
immediately before the pulse leaves the zone. This
happens when the attractive part of the pulse
(cf. Fig. 1e) just has passed into the lower 30 ML.
Finally, Fig. 8 shows what happens when the
pulse impinges on the boundary zone presented in
this paper. The strong decrease of the energy at 1.1
ps proves that this boundary zone essentially acts
like the ideal boundary. However, several differences are seen in detail. When the pulse enters the
boundary zone, at first atoms are decelerated due
to the action of the boundary force, Eq. (4). Then,
at t ffi 1 ps, atoms are accelerated, in close analogy
to what happens at a free boundary, Fig. 6.
However, while there the crystal relaxed almost
completely to Epot ¼ 0, Epot increases in the boundary zone, since atom motion is hindered, and
hence repulsive forces dominate. Also after pulse
passage, t > 1 ps, the energies show a temporal
dependence different from that of the ideal boundary, Fig. 7; and in particular not all energy leaves
the crystal.
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